Find My says to sign in even though I'm signed in

This identical issue was raised in 2020: Find My says to sign in even though I'm s… - Apple Community

The solution proposed there — create a new user account — seems impractical and nonsensical.

A few weeks ago I started using a new MacBook Pro (M1 Max, 16-inch, 2021) running macOS Monterey. Everything seems to be working fine except Find My. On my iPhone (13 from last fall) it works fine, I see the location of my family and their devices, except for my laptop.

On the laptop, the People tab of Find My says “Sign In. Start sharing your location with friends or family members.” Below that is a button saying System Preferences which takes me to the Apple ID tab. That shows me signed into Apple ID. In the iCloud sub-tab, the check box for Find My Mac is checked. I have tried unchecking and rechecking that, but it had no effect.